Properties

Degree 12
Conductor $ 2^{18} \cdot 3^{6} \cdot 5^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 4-s + 2·8-s + 21·9-s − 6·12-s + 3·16-s − 12·24-s − 56·27-s − 12·31-s + 4·32-s + 21·36-s − 8·37-s − 20·41-s + 8·43-s − 18·48-s + 6·49-s − 12·53-s + 64-s − 24·67-s − 8·71-s + 42·72-s − 36·79-s + 126·81-s − 32·83-s + 28·89-s + 72·93-s − 24·96-s + ⋯
L(s)  = 1  − 3.46·3-s + 1/2·4-s + 0.707·8-s + 7·9-s − 1.73·12-s + 3/4·16-s − 2.44·24-s − 10.7·27-s − 2.15·31-s + 0.707·32-s + 7/2·36-s − 1.31·37-s − 3.12·41-s + 1.21·43-s − 2.59·48-s + 6/7·49-s − 1.64·53-s + 1/8·64-s − 2.93·67-s − 0.949·71-s + 4.94·72-s − 4.05·79-s + 14·81-s − 3.51·83-s + 2.96·89-s + 7.46·93-s − 2.44·96-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(12\)
\( N \)  =  \(2^{18} \cdot 3^{6} \cdot 5^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{600} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((12,\ 2^{18} \cdot 3^{6} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(0.159819\)
\(L(\frac12)\)  \(\approx\)  \(0.159819\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 12. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 11.
$p$$F_p(T)$
bad2 \( 1 - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6} \)
3 \( ( 1 + T )^{6} \)
5 \( 1 \)
good7 \( 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12} \)
11 \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \)
13 \( ( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2} \)
17 \( 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} \)
19 \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \)
23 \( 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \)
29 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \)
31 \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
37 \( ( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
41 \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
43 \( ( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
47 \( 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \)
53 \( ( 1 + 2 T + p T^{2} )^{6} \)
59 \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \)
61 \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \)
67 \( ( 1 + 4 T + p T^{2} )^{6} \)
71 \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
73 \( ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3} \)
79 \( ( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 317 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
83 \( ( 1 + 16 T + 265 T^{2} + 2400 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
89 \( ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \)
97 \( 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 p^{4} T^{10} + p^{6} T^{12} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−5.62127773430141346814980798220, −5.57605520652319942218035698053, −5.55728717117692810265656600414, −5.50911098886993741796080005905, −4.89450136833995656433304553409, −4.89163429170202900306328328012, −4.77537189873276606493055123896, −4.63920355519819174369012170356, −4.51384057681432099411117707924, −4.18420043824646641370016337398, −4.05350888111005769883656762104, −3.93153066843160318929114544702, −3.61138327812209258684653296657, −3.33983476215168578230037569483, −3.29705382881241156709347973515, −2.86838385467434311690789796059, −2.84517853161744640264327861742, −2.37782268559277801117909772703, −2.02945077535826425175472285014, −1.59766349238880785080994304301, −1.59591393022744840411065286057, −1.41435310844905832860546651901, −1.33378189332377377405769252178, −0.54388433427110706472065004062, −0.15903604181634495935897673885, 0.15903604181634495935897673885, 0.54388433427110706472065004062, 1.33378189332377377405769252178, 1.41435310844905832860546651901, 1.59591393022744840411065286057, 1.59766349238880785080994304301, 2.02945077535826425175472285014, 2.37782268559277801117909772703, 2.84517853161744640264327861742, 2.86838385467434311690789796059, 3.29705382881241156709347973515, 3.33983476215168578230037569483, 3.61138327812209258684653296657, 3.93153066843160318929114544702, 4.05350888111005769883656762104, 4.18420043824646641370016337398, 4.51384057681432099411117707924, 4.63920355519819174369012170356, 4.77537189873276606493055123896, 4.89163429170202900306328328012, 4.89450136833995656433304553409, 5.50911098886993741796080005905, 5.55728717117692810265656600414, 5.57605520652319942218035698053, 5.62127773430141346814980798220

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.